Scalar-Time Hamiltonian Resonance Dynamics in Multi-Body Orbital Systems: A Canonical Time–Scalar Field Theory Framework for Transit Timing Variations

We develop a canonical Time–Scalar Field Theory (TSFT) treatment of resonant multi-body orbital dynamics and apply it to transit timing variations (TTVs) in compact planetary systems. Unlike earlier formulations that introduced resonance corrections at the level of an effective ansatz, the present work derives the orbital framework directly from the established TSFT canonical structure: scalar-time Hamiltonian flow, action reparameterization in Θ, and the weak-field gravitational identification Θ = Φ/c^2. In this formulation, orbital evolution is not taken to occur fundamentally in coordinate time t, but along the scalar-time field Θ, with ordinary time recovered as a reparameterized projection. This yields a natural description of resonant phase-locking as a canonical perturbation problem under scalar-time flow. Starting from the archived TSFT equationsdO/dΘ= α{O,H} + ∂O/∂Θ, α ≡ dt/dΘ, a = (−c^2)(∇Θ), we derive the effective resonance dynamics of coupled orbital phases and show that transit timing offsets arise as observable projections of scalar-time phase drift. The resulting framework recovers the qualitative structure of known resonant timing behavior while recasting the classical three-body problem as a synchronization problem in the underlying time field rather than as purely geometric chaos in coordinate space. We further obtain a first-order TTV scaling relation from the canonicalperturbation structure and discuss its interpretation in systems such as Kepler-9, TRAPPIST-1, and TOI-270. The main contribution of this paper is therefore not a claim of closed-form global solution of the classical three-body problem, but a mathematically sharper statement: resonant multi-body timing structure can be modeled in TSFT as Hamiltonian phase evolution on a scalar-time manifold, with observed timing anomalies emerging from projection between Θ-flow and clock-time observables. This places prior TSFT orbital ideas on a more rigorous internal footing and clarifies their relationship to standard Hamiltonian perturbation theory.